Abstract
The spin of the electron is nowadays replacing the charge as basic carrier of information not only in spintronics applications, but also in the emerging field of quantum information. Topological quantum materials, where spinmomentum locking is believed to lead to particularly long spin lifetimes, are regarded as a promising platform for such applications. However, spinorbit coupling, that is essential to all topological matter, at the same time gives rise to spin mixing and decoherence as a major obstacle for quantum computing. Here, we give experimental evidence that hotspots of spinmixing and spinconserving contributions of the spinorbit operator coexist in an archetypal topological Dirac metal, and that these hot spots can have a strongly anisotropic distribution of their respective wave vectors with respect to the spin quantization direction. Our results can be understood within a theory that takes into account the decomposition of the spinorbit Hamiltonian into spinconserving and spinflip terms, contributing to a better understanding of quantum decoherence in topological materials, in general.
Introduction
In spintronic devices, the spin of the electron becomes the main carrier of information, providing a link between magnetism and conventional chargebased electronics. In this approach, spinup (↑) and spindown (↓) states of the electron spin serve as a classical bit to represent and store binary information^{1,2}. On the microscopic level, however, the electron spin by its nature has a quantum mechanical origin. As a consequence, the statistical interpretation of quantum mechanics provides possibilities to find the eigenvalue \(\frac{\hslash }{2}\) and \(\frac{\hslash }{2}\) for spins pointing up and down along a given quantization axis. In particular, any given spin state can be represented as a superposition,
of these two states. Quantum physics therefore allows information to be represented and processed in a significantly different manner, beyond the classical interpretation of spintronics. In this picture of spinbased quantum computing, the coherent superposition of two distinct spin states represents a single quantum bit, a socalled qubit^{2,3}.
A fundamental mechanism that allows the control of the electron spin in condensed matter is provided by spin–orbit coupling (SOC). Due to SOC, the eigenstates of the Hamiltonian are no longer wave functions with a pure spin character, but a superposition of up and down states. This manifests in a manifold of phenomena. On the one side, SOC is the source of spinmomentum locking found in the electronic structure of topological quantum materials, leading to particularly long spin lifetimes. By contrast, the mixture of spinup and spindown states due to SOC enables that scattering events will flip the spin of an electron with a certain probability. For instance, the Elliott–Yafet spinrelaxation mechanism, being driven by SOC, represents a major source of spin decoherence^{2,4,5}.
To understand this apparent discrepancy, the SOC operator can be divided into a spinconserving \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \uparrow }\) and a spinflip \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \downarrow }\) part:
For a fixed direction of the spin quantization axis (SQA), these two contributions correspond to states with a pure spin character and fully spinmixed states, respectively. Here, ξ is the strength of SOC, L and S are the orbital and spin angular momentum operators, \({L}_{\hat{s}}={{{{{{{\bf{L}}}}}}}}\cdot \hat{s}\), \({S}_{\hat{s}}={{{{{{{\bf{S}}}}}}}}\cdot \hat{s}\), and \(\hat{s}\) is a unit vector in the direction of the SQA. As the spin–orbit operator is a nonlocal operator, it depends on the energy and wave vector of the respective states. In crystalline solids, thus, the electronic structure can have regions in the Brillouin zone (BZ) of particularly large spin relaxation. Such regions have, for instance, direct consequences in the interpretation of ultrafast dynamics of spindependent properties^{1,5,6,7,8,9,10}. A direct experimental access to the contrasting contributions of SOC in the electronic structure has so far remained elusive. Yet, such information would help to identify spots of long/short spin lifetime and might pave the way towards a precise control of spin states via ondemand tailoring of the SOC.
Here we used spin and momentumresolved photoelectron spectroscopy to map the SOC spinmixing parameter in the Fermi surface (FS) of the prototypical heavy transition metal tungsten (W). Tungsten has the highest charge to spincurrent conversion efficiency among all 5d transition metals, suggesting a significant role of SOC in its FS topology^{11,12,13,14,15}. We have investigated the twofold rotational symmetric (C_{2v}) W(110) surface. In reminiscence of a crystalline topological insulator, the mirror symmetry of W(110) leads to several dorbitalderived surface states with a helical spin texture^{16,17,18,19}. The W(110) surface thus represents a paradigmatic model system to understand the nonlocal distribution of spinconserving and spinmixing parts of the SOC operator in the electronic structure. Our experimental and theoretical results give direct evidence that spin mixing in the W(110) surface becomes strongly anisotropic, with a pronounced nonuniform distribution of spinmixing hotspots in the FS. Our results have important consequences for spintronics and understanding the efficiency of spintorques and spincurrent generation, as well as spincoherence effects in quantum materials.
Results
Multiple dderived Dirac fermions in W(110)
Figure 1a presents the layout of our spinresolved momentum microscopy experiments. Photoelectrons are collected simultaneously over the full solid angle above the sample, allowing the efficient measurement of the spinresolved spectral function over the complete surface BZ (SBZ) (see ‘Methods’)^{20,21,22}. The momentum image shows the electronic states at the Fermi energy E_{F}.
Figure 1b–d show sections through the electronic structure between E_{F} and E_{F} − 1.7 eV. The W(110) surface is known to host in total three gapless points in the SBZ forming Dirac cones with linear dispersion: one appears at the centre and two are away from the highsymmetry points of the SBZ^{17,18,19}. In our measurements, we can clearly observe all three Dirac cones in the SBZ. Their positions are summarized in Fig. 1e. These three Dirac cones can be classified into two sets with distinct group velocities: one at the \(\overline{{{\Gamma }}}\) point and two at k_{y} = ±0.4 Å^{−1}. They are located inside a spin–orbitinduced bulk band gap that resembles the formation of the Diraccone surface band inside the fundamental gap of a topological insulator^{17}. The linear bands cross at the \(\overline{{{\Gamma }}}\) point at E_{B} = 1.25 eV (see Fig. 1b, c). At k_{y} = ±0.4 Å^{−1}, the Dirac point lies at E_{B} = 0.75 eV, as shown in Fig. 1d. The presence of an odd number of gapless Dirac cones indicates a nontrivial topological character. Tungsten has been therefore classified as a dorbitalderived topological crystalline material^{16} where the Dirac cones are protected by the mirror symmetry of the (110) surface.
Full spin texture
Because of the combination of the twofold surface symmetry and timereversal symmetry, the outofplane component of spin polarization vanishes at all kpoints in W(110)^{23}, i.e., P_{z} = 0. A negligible outofplane spin polarization is further confirmed by our firstprinciple calculations. As a result, the initialstate spin polarization is purely inplane, with components P_{x} and P_{y}. During the photoemission by linearly polarized light, additional contributions to the spin polarization may arise due to the lowered symmetry by the incidence direction of the light, as discussed in detail in refs. ^{24,25}. Here we chose a geometry where the SQA lies within the optical plane, such that these effects are minimized (see ‘Methods’) and the measured spin polarization is expected to closely reflect the W(110) FS spin texture. In particular, when the W(110) surface is excited with linearly polarized light, a purely inplane spin polarization has been previously reported, whereas a nonzero P_{z} component can be obtained using circularly polarized light^{26}.
Figure 2a shows the measured spinresolved FS contour of W(110) for a vertical P_{y} inplane polarization, the same spin direction as discussed in Fig. 1. The measured FS contour agrees well with previous results that have probed P_{y}^{18}. In particular, we find a high polarization reaching P_{y} values up to ±0.7 in the pronounced crossshaped contour. These states can be related to dorbitalderived surface resonances^{27}. The spinresolved measurement reveals that a Rashbalike spin splitting is found around the \(\overline{{{\Gamma }}}\) point (A), whereas the vertical contour of the FS (B) at k_{y} = ±0.4 Å^{−1} merge into the offcentred Dirac cone observed in Fig. 1d. The ellipseshaped contours around the \(\overline{S}\) points are surface states that exhibit a small Rashbalike spin splitting.
A significant change of the spin polarization is found for the P_{x} component in Fig. 2b: in general, P_{x} changes the sign upon the mirror operation k_{y} → −k_{y}, but is unchanged upon k_{x} → −k_{x}. This reflects the fact that the spin angular momentum described by pseudovectors fulfils timereversal and mirror symmetries. The overall magnitude of the measured P_{x} only reaches values of ±0.5 and is significantly reduced compared to P_{y}. In particular, P_{x} vanishes along the vertical sections of the main FS contour (C) at k_{y} ≈ ±0.4 Å^{−1} and only small polarizations are observed along the horizontal sections (D). This provides already a qualitative picture of the inplane spin texture: the spin is locally aligned tangential to the FS contour, in reminiscence of the spin warping observed in some topological insulators rather than being locked normal to the kvector as in a regular Rashba model.
Figure 2c, d show the theoretical spinresolved FS for P_{y} and P_{x}, respectively. The calculation reproduces the main details of the experiment, in particular the P_{x} and P_{y} spin polarizations along the crossshaped FS contour. As in the experiment, the maximum polarization is observed in the P_{y} component along the vertical sections in vicinity of the offcentred Dirac cone (E), whereas P_{x} vanishes for these states (F). Furthermore, we clearly observe the Rashbasplitting of the surface states around the \(\overline{S}\) points. The calculation further confirms the experimental observation that both P_{x} and P_{y}, at the same time, show a strongly reduced polarization along the vertical k_{x} = 0 mirror plane of W(110). The quantitative overall agreement between theory and experiments confirms that our measured results mainly represent the initialstate spin polarization.
Figure 3a shows the spin texture over the whole SBZ. As discussed above, in the outer contour of the crosslike feature, spinmomentum locking leads to a counterclockwise rotation around the BZ centre. At larger binding energies, these states converge into the Diraccone states. The two nearly degenerate ellipseshaped surface states around \(\overline{S}\) are very close to each other. The Rashbalike opposite spin chirality explains the low observed spin polarization for P_{x} and P_{y} (see Fig. 2a, b).
Anisotropic quantum spin mixing
Due to the helical spin texture, one would anticipate a similarly high degree of spin polarization for the P_{x} and P_{y} momentum maps. For instance, this would be the expected case for the spin texture of an archetypal Rashba surface state with isotropic freeelectronlike bands. Such model systems are the Lgap surface states on the (111) faces of noble metals like Au(111)^{20}. In contrast, we find a pronounced dependence of the observed spin polarization on the choice of the SQA: the overall spin polarization P_{x} is substantially suppressed, compared to P_{y}. To understand the underlying mechanism of how the choice of the SQA of an incoming conduction electron affects the degree of spin polarization, we carried out ab initio calculations^{5} to analyse the momentum distribution of the spinmixing parameter.
Figure 3b, c show the theoretical momentumresolved spinmixing parameters \({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) (see ‘Methods’)^{5} of the W(110) FS when the SQA points along the y and x axes, respectively. A fully spinmixed state (\({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) = \(\frac{1}{2}\)) is represented by a superposition of \(\left\uparrow \right\rangle\) and \(\left\downarrow \right\rangle\) spin states with a probability of 0.5. As a spinresolved measurement integrates over an ensemble of many observed electrons, spinmixing hot areas (red colours in Fig. 3) consequently appear with no effective spin polarization in the experiment. Likewise, a pure spin state corresponds to \({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) = 0 and results in a high spin polarization (positive or negative) along the respective SQA. The measured spinpolarization maps in Fig. 2a, b thus provide a direct experimental access to the distribution of the spinmixing parameter in the SBZ.
By rotating the SQA from the y to the xaxis, the spin mixing varies considerably. With the SQA∥y, spin mixing is relatively uniform over the FS. A few hotspots of large spin mixing are located at places of large band curvature and along the vertical k_{x} = 0 plane. This picture changes dramatically when the SQA∥x. In the latter case, full spin mixing is prominent in large areas of the SBZ. In these regions, the Bloch wave functions become a superposition of spinup and spindown states of the same magnitude.
This result is in perfect agreement with our experimental observations in Fig. 2b, where an overall low spin polarization is found. If an electron is scattered into such states, there is a high probability that the spin flips. They thus represent a source of decoherence, degrading the ability of qubits and spintronics to retain encoded information. As a consequence, the strong spin mixing gives rise to hot areas of enhanced spin flip when the SQA points along the x, but not when the SQA points along the ydirection.
Discussion
Due to SOC, the spin and orbital character of the Bloch states depends strongly on the spin orientation with respect to the crystal axes. This leads to drastic changes in the spinconserving \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \uparrow }\) and a spinmixing \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \downarrow }\) part of the electronic states with respect to the choice of the SQA. We note that the total SOC term (Eq. (2)) is a scalar product, which is independent of the SQA. This representation thus allows us to describe the anisotropic distribution of spin mixing in the entire BZ. In particular, a large \(\xi {({L}_{\hat{s}}{S}_{\hat{s}})}^{\uparrow \downarrow }\) indicates a mixing of spin states due to the presence of SOC.
The distribution of the spin mixing in the BZ has immediate consequences for spin dynamics and spinrelated properties^{1,5,6,7,8,9,10}. For instance, in regions of the BZ where the spinconserving part of SOC is dominant, the electron spin is protected from dissipative effects, giving rise to relatively long spin lifetimes. The spin information of these states is thus preserved, being useful when a long spin lifetime is needed in spintronic devices and quantum information applications. On the other hand, regions with an enhanced spinmixing will have a major contribution to spinrelaxation processes and ultrafast spin dynamics.
Although the eigenenergies of the Bloch states are independent of the SQA, the choice of the SQA can alter the momentum distribution of spinconserving and spinmixing contributions. For simplicity, spin relaxation as it is described by the Elliott–Yafet mechanism^{2,4} does not take into account these individual relaxation hotspots in the BZ. Nevertheless, it has been shown that the macroscopic Elliott–Yafet relaxation parameter can be obtained directly by integrating the wavevectordependent spinrelaxation parameter from Fig. 3b, c over the BZ^{5}. The wave vector of the respective electron states, however, becomes of importance in singlecrystalline materials and junctions, where tunnelling currents, spindependent excitations, and spin–orbit torque are selectively driven by small areas in the FS of the materials^{1,5,14,28,29}. Spinconserving and spinmixing contributions thus are dominated by hotspots in the FS and can be exploited by appropriately designed devices.
Our findings show that the spin mixing of the Bloch states in a nonmagnetic material can change drastically when the spin quantization direction varies. As a consequence, the spinmixing parameter becomes strongly anisotropic. The Dirac surface states, marked by H, serve as the main source of anisotropy, with the spinmixing parameter changing by >50% between SQA∣∣x and SQA∣∣y. This result can be attributed to strong spinmomentum locking perpendicular to the k_{x} momentum direction. In contrast, the states marked by L show almost no anisotropy. A large spin mixing for both SQAs is here mediated by a strong overlap with the projected bulk bands^{27}.
The spinmixing parameter introduced here is a direct consequence of the superposition principle of quantum mechanics. Our observations confirm experimentally the possibility to visualize the effect of a superposition of quantum mechanical wave functions, namely the spinup and spindown states that form a mixed spin state due to a strong SOC. We demonstrate the potential scope of spinmomentum locking for quantum spinmixing manipulation in Dirac materials. Our findings are widely applicable in abundant crystals with SOC and offer promising perspectives to finetune the quantum (de)coherence within one material by changing the spin polarization of a conduction electron in a spintronic device.
Methods
Spinresolved momentum microscopy
Spin and momentumresolved photoelectron spectroscopy experiments were carried out at the NanoESCA beamline^{30} of the Elettra synchrotron in Trieste (Italy), using ppolarized photons at the photon energy hν = 50 eV. All measurements were performed while keeping the sample at a temperature of 130 K. Photoelectrons emitted into the complete solid angle above the sample surface were collected using a momentum microscope^{20,22}. The momentum microscope directly forms an image of the distribution of photoelectrons as a function of the lateral crystal momentum (k_{x}, k_{y}) that is recorded on an imaging detector^{20,22}.
An imaging spin filter based on the spindependent reflection of lowenergy electrons at a W(100) single crystal^{31} allows the simultaneous measurement of the spin polarization of photoelectrons in the whole SBZ. Images were recorded after reflection at a scattering energy at 26.5 and 30.5 eV, which changes the spin sensitivity S of the detector between 42% and 5%, respectively. From these images, the spin polarization at every (k_{x}, k_{y}) point in a momentum map at certain binding energies (e.g., Figs. 1 and 2 in the main study) is derived^{21,31,32}. As a dorbitalderived topological metal, sizable photoemission intensities of the Dirac states are mainly observed by excitation with ppolarized light^{33}. As verified by our firstprinciples calculations (see Fig. 2c, d), our measurements using ppolarized light reproduce well the groundstate spin polarization, such that pronounced effects by optical orientation, which are observed for porbital topological materials, only play a minor role^{34,35}.
The spinresolved momentum microscopy measurements were performed along the quantization axis of the spin filter, being the same direction as \(\overline{{{\Gamma }}}\overline{{{{{{{{\rm{N}}}}}}}}}\), k_{y} of W(110). In order to measure the transverse component of the spin polarization parallel to the k_{x} (\(\overline{{{\Gamma }}}\overline{{{{{{{{\rm{H}}}}}}}}}\)) direction of W(110), we rotated the sample around 90°. The k_{x} direction of the sample then is parallel to the quantization axis of the spin filter (see Fig. 2 of the main manuscript). In this way, the SQAs for the measurements of the P_{x} and P_{y} components always lie within the optical plane, such that timereversal and mirror symmetries of the spin polarization are preserved.
The W(100) crystal of the spin filter was prepared by following the same procedure as described below for W(110). This standard procedure is known to lead to clean tungsten surfaces^{31,32}. The spinresolved photoemission momentum maps for each constant energy contour, as shown in Fig. 1a, were measured only for 6 min. This allows to collect spinresolved data in a wide range of binding energies, E_{B}. The complete 3D (k_{x}, k_{y}, E_{B}) information of the spinresolved spectral function is obtained within a 2 h measurement time. Despite many attempts, it has not been possible so far to achieve a comparably detailed information by conventional spinresolved photoemission spectroscopy^{36}.
Cleaning procedure for a tungsten single crystal
The procedure for cleaning W(110) consists of two steps: (i) cycles of lowtemperature flash heating (T ~ 1200 K) at an oxygen partial pressure of \({P}_{{{{{{{{{\rm{O}}}}}}}}}_{2}}=5\times 1{0}^{8}\) mbar to remove the carbon from the surface and (ii) a single hightemperature flash (T ~ 2400 K) to remove the oxide layer^{32}. The cleanliness of the W(110) surface was checked by lowenergy electron diffraction and Auger electron spectroscopy.
Firstprinciples calculations
Using the fullpotential linearized augmentedplanewave method as implemented in the FLEUR code (http://www.flapw.de), we performed density functional theory calculations of 7 layers of W(110). We adopted the structural parameters given by Heide et al.^{37}. Exchange and correlation effects were treated within the localdensity approximation (LDA)^{38}. The spin–orbit interaction was included selfconsistently and the spin quantization direction was chosen with respect to \(\overline{{{\Gamma }}}\overline{{{{{{{{\rm{N}}}}}}}}}\) and \(\overline{{{\Gamma }}}\overline{{{{{{{{\rm{H}}}}}}}}}\) directions of W(110). The planewave cutoff was chosen as \({k}_{\max }=4.2\ {a}_{0}^{1}\) with a_{0} as Bohr’s radius and the muffintin radii set to 2.5 a_{0}.
To determine the FS with high precision, we calculated the eigenspectrum on a dense uniform mesh of 256 × 256 kpoints in the full BZ. The Fermilevel crossings were then obtained by triangulation. To improve the agreement with experiment, we shifted the position of the Fermi level in our calculations downwards by 70 meV as compared to the theoretical value for the Fermi energy.
Our investigation for the momentum distribution of the spinmixing parameter is based on density functional calculation within the LDA^{39}. We employ the fullpotential Korringa–Kohn–Rostoker Green function method^{40} with exact treatment of the atomic cell shapes^{41,42}. After a selfconsistent fullpotential calculation performed within the scalarrelativistic approximation, SOC is added when calculating the FS properties^{43}. The spinmixing parameter is given by
Wave functions are chosen in a way that the spin \(\langle {S}_{\hat{s}}\rangle\) along the SQA \(\hat{s}\) is maximized. \({b}_{{{{{{{{\bf{b}}}}}}}}\hat{s}}^{2}\) can then obtain values between 0 (pure spin state) and \(\frac{1}{2}\) (fully mixed spin state). Usually, the Bloch states are of nearly pure spin character. However, at special spin flip hotspots in the BZ spin mixing may increase significantly up to 1/2, which corresponds to the case of fully spinmixed states.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Sinova, J. & Žutić, I. New moves of the spintronics tango. Nat. Mater. 11, 368 (2012).
Awschalom, D. D., Bassett, L. C., Dzurak, A. S., Hu, E. L. & Petta, J. R. Quantum spintronics: engineering and manipulating atomlike spins in semiconductors. Science 339, 1174–1179 (2013).
Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Murakami, S., Nagaosa, N. & Zhang, S.C. Dissipationless quantum spin current at room temperature. Science 301, 1348–1351 (2003).
Zimmermann, B. et al. Anisotropy of spin relaxation in metals. Phys. Rev. Lett. 109, 236603 (2012).
Fabian, J. & Das Sarma, S. Spin relaxation of conduction electrons in polyvalent metals: theory and a realistic calculation. Phys. Rev. Lett. 81, 5624–5627 (1998).
Soumyanarayanan, A., Reyren, N., Fert, A. & Panagopoulos, C. Emergent phenomena induced by spin–orbit coupling at surfaces and interfaces. Nature 539, 509 (2016).
Kurebayashi, H. et al. An antidamping spinorbit torque originating from the Berry curvature. Nat. Nanotechnol. 9, 211– (2014).
Seifert, T. S. et al. Femtosecond formation dynamics of the spin seebeck effect revealed by terahertz spectroscopy. Nat. Commun. 9, 2899 (2018).
Siegrist, F. et al. Lightwave dynamic control of magnetism. Nature 571, 240–244 (2019).
Wang, M. et al. Currentinduced magnetization switching in atomthick tungsten engineered perpendicular magnetic tunnel junctions with large tunnel magnetoresistance. Nat. Commun. 9, 671– (2018).
Sheng, P. et al. The spin Nernst effect in tungsten. Sci. Adv. 3, e1701503 (2017).
Pai, C.F. et al. Spin transfer torque devices utilizing the giant spin hall effect of tungsten. Appl. Phys. Lett. 101, 122404 (2012).
Cho, S., Baek, S.hC., Lee, K.D., Jo, Y. & Park, B.G. Large spin hall magnetoresistance and its correlation to the spinorbit torque in W/CoFeB/MgO structures. Sci. Rep. 5, 14668 (2015).
Kim, D.J. et al. Observation of transverse spin nernst magnetoresistance induced by thermal spin current in ferromagnet/nonmagnet bilayers. Nat. Commun. 8, 1400 (2017).
Zhang, T. et al. Catalogue of topological electronic materials. Nature 566, 475 (2019).
Varykhalov, A. et al. Tilted dirac cone on W(110) protected by mirror symmetry. Phys. Rev. B 95, 245421 (2017).
Kutnyakhov, D. et al. Spin texture of timereversal symmetry invariant surface states on W(110). Sci. Rep. 6, 29394 (2016).
Miyamoto, K. et al. Spinpolarized Diracconelike surface state with d character at W(110). Phys. Rev. Lett. 108, 066808 (2012).
Tusche, C., Krasyuk, A. & Kirschner, J. Spin resolved bandstructure imaging with a high resolution momentum microscope. Ultramicroscopy 159, 520–529 (2015).
Tusche, C. et al. Nonlocal electron correlations in an itinerant ferromagnet. Nat. Commun. 9, 3727 (2018).
Tusche, C., Chen, Y.J., Schneider, C. M. & Kirschner, J. Imaging properties of hemispherical electrostatic energy analyzers for high resolution momentum microscopy. Ultramicroscopy 206, 112815 (2019).
Mirhosseini, H., Giebels, F., Gollisch, H., Henk, J. & Feder, R. Ab initio spinresolved photoemission and electron pair emission from a Diractype surface state in W(110). N. J. Phys. 15, 095017 (2013).
Heinzmann, U. & Dil, J. H. Spin–orbitinduced photoelectron spin polarization in angleresolved photoemission from both atomic and condensed matter targets. J. Phys. Condens. Matter 24, 173001 (2012).
Osterwalder, J. Can spinpolarized photoemission measure spin properties in condensed matter? J. Phys. Condens. Matter 24, 171001 (2012).
Miyamoto, K., Wortelen, H., Okuda, T., Henk, J. & Donath, M. Circularpolarizedlightinduced spin polarization characterized for the Diraccone surface state at W(110) with C_{2v} symmetry. Sci. Rep. 8, 10440 (2018).
Elmers, H. J. et al. Hosting of surface states in spin–orbit induced projected bulk band gaps of W(110) and Ir(111). J. Phys. Condens. Matter 29, 255001 (2017).
Tusche, C. et al. Oxygeninduced symmetrization and structural coherency in Fe/MgO/Fe(001) magnetic tunnel junctions. Phys. Rev. Lett. 95, 176101 (2005).
Tusche, C. et al. Tusche et al. reply. Phys. Rev. Lett. 96, 119602 (2006).
Wiemann, C. et al. A new nanospectroscopy tool with synchrotron radiation: Nanoesca@Elettra. eJ. Surf. Sci. Nanotechnol. 9, 395–399 (2011).
Tusche, C. et al. Spin resolved photoelectron microscopy using a twodimensional spinpolarizing electron mirror. Appl. Phys. Lett. 99, 032505 (2011).
Tusche, C. et al. Quantitative spin polarization analysis in photoelectron emission microscopy with an imaging spin filter. Ultramicroscopy 130, 70–76 (2013).
Miyamoto, K. et al. Orbitalsymmetryselective spin characterization of Diracconelike state on W(110). Phys. Rev. B 93, 161403(R) (2016).
Bentmann, H. et al. Profiling spin and orbital texture of a topological insulator in full momentum space. Phys. Rev. B 103, L161107 (2021).
Maaß, H. et al. Spintexture inversion in the giant Rashba semiconductor BiTeI. Nat. Commun. 7, 11621 (2016).
Suga, S. & Tusche, C. Photoelectron spectroscopy in a wide hν region from 6 eV to 8 keV with full momentum and spin resolution. J. Electron Spectrosc. Relat. Phenom. 200, 119–142 (2015).
Heide, M., Bihlmayer, G. & Blügel, S. DzyaloshinskiiMoriya interaction accounting for the orientation of magnetic domains in ultrathin films: Fe/W(110). Phys. Rev. B 78, 140403 (2008).
Janak, J. F., Williams, A. R. & Moruzzi, V. L. Local exchangecorrelation potentials and the Fermi surface of copper. Phys. Rev. B 6, 4367–4370 (1972).
Vosko, S. H., Wilk, L. & Nusair, M. Accurate spindependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys. 58, 1200–1211 (1980).
Papanikolaou, N., Zeller, R. & Dederichs, P. H. Conceptual improvements of the KKR method. J. Phys. Condens. Matter 14, 2799–2823 (2002).
Stefanou, N. & Zeller, R. Calculation of shapetruncation functions for voronoi polyhedra. J. Phys. Condens. Matter 3, 7599–7606 (1991).
Stefanou, N., Akai, H. & Zeller, R. An efficient numerical method to calculate shape truncation functions for wignerseitz atomic polyhedra. Comput. Phys. Commun. 60, 231–238 (1990).
Heers, S. Effect of spinorbit scattering on transport properties of lowdimensional dilute alloys. PhD thesis, RWTH Aachen Univ. (2011).
Acknowledgements
Y.J.C. and C.T. thank the staff of Elettra for their help and hospitality during their visit in Trieste, and beamline staff M. Jugovac, G. Zamborlini, V. Feyer (PGI6, FZJülich), and T. O. Menteş (Elettra) for their assistance during the experiment and providing the W(110) crystal. We are indebted to J.P. Hanke and Y. Mokrousov for discussions. Y.J.C., C.T., and C.M.S. gratefully acknowledge funding by the BMBF (Grant number 05K19PGA). The authors gratefully acknowledge the computing time granted through JARA on the supercomputer JURECA at Forschungszentrum Jülich.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Contributions
Y.J.C. and C.T. carried out the experiment and analyzed the experimental data. M.H. and B.Z. carried out firstprinciple calculations and analyzed the theoretical data with assistance by G.B. Y.J.C. drafted the manuscript with assistance by C.T. C.T. designed and coordinated the research together with C.M.S. and S.B. All authors discussed the results and contributed to the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Communications Physics thanks Koji Miyamoto and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, YJ., Hoffmann, M., Zimmermann, B. et al. Quantum spin mixing in Dirac materials. Commun Phys 4, 179 (2021). https://doi.org/10.1038/s42005021006825
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005021006825
This article is cited by

Momentumselective orbital hybridisation
Nature Communications (2022)

Spanning Fermi arcs in a twodimensional magnet
Nature Communications (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.